In Time-Division Synchronous Code-Division Multiple-Access (TD-SCDMA) and Long-Term Evolution (LTE) systems, an adaptive antenna array is used to increase the system coverage and capacity, etc., which is a group of antenna elements located very close to each other (e.g. half of wave length of the carrier frequency). The antenna array can send a target signal directed to a target user with low interference to other users by means of low transmitting power in the direction of other users. To make sure the signal directed to the expected user, the delay, phase and amplitude of each antenna channel have to keep identical during the signal transmitting or receiving. The beamforming technology is useful in sending carefully calculated signal on each antenna to make the transmitted signal in the air be directed to the target user.
In reality, the delay, phase and amplitude of each antenna channel cannot be identical naturally, because of the temperature, thermal noise, inconsistency of components, etc. The above uncertainties lead to random delay, random phase and amplitude of each antenna channel.
However, for many applications of an adaptive antenna array, it is required that all elements of the antenna array have equal gain and phase characteristics, so calibrating the individual elements/TX (RX) path is of utmost importance. In the 3GPP LTE-TDD standard, an OFDM system is adopted for the downlink with multiple antennae. Beamforming is also one of the critical features for the system. To obtain a beamforming gain with symmetrical downlink and uplink wireless channels, the gain and phase characteristics of multiple antennae should remain identical.
In the RF part of a real system, the amplitude, phase and delay of various antenna channels (including digital upper converter (DUC), digital down converter (DDC), digital to analogue converter (DAC), and analogue to digital converter (ADC) etc) are always different from channel to channel in both the downlink direction and the uplink direction, as shown in FIG. 1. In the downlink direction, to obtain the beamforming gain, it is important to calibrate the antenna RF channels to make the channel response of each channel the same. Similarly, to use the estimated uplink wireless channel to generate a beamforming vector, the channel response of the RF part in the uplink direction should also be the same. Therefore a calibration in the uplink and downlink is needed. To calibrate the delay, amplitude and phase differences of the channel, a delay difference are estimated first. Thus it becomes necessary to determine how to estimate and calibrate the delay difference of antenna RF channels.
In the OFDM system, besides the amplitude and the phase difference of the antennas, the time delay difference also adversely affects the consistency in channel response. Moreover, the time delay difference tends to more significantly affect the higher frequency band and the lower frequency band in the signal bandwidth, since the delay difference results in phase difference in the frequency domain. Thus, the higher/lower the frequency band is, the larger the phase difference will be.
In the OFDM system, the conversion from the time domain reference data x=[x0, x1, . . . , xN−1] to frequency domain data y=[y0, y1, . . . , yN−1] is expressed in the following Fast Fourier Transform (FFT) operation:
            y      k        =                  ∑                  n          =          0                          N          -          1                    ⁢                        x          n                ⁢                  exp          ⁡                      (                                          -                j                            ×                                                2                  ⁢                  π                                N                            ×              n              ×              k                        )                                ,          ⁢      0    ≤    k    ≤          N      -      1.      
If there is an amplitude, phase or delay error, the frequency domain signal should be expressed as follows:
      r    =          [                        r          0                ,                  r          1                ,        …        ⁢                                  ,                  r                      N            -            1                              ]                                                r            k                    =                    ⁢                                    h              ×                                                ∑                                      n                    =                    0                                                        N                    -                    1                                                  ⁢                                                      x                    n                                    ⁢                                      exp                    ⁡                                          (                                                                        -                          j                                                ×                                                                              2                            ⁢                            π                                                    N                                                ×                                                  (                                                      n                            +                                                          Δ                              ⁢                                                                                                                          ⁢                              t                              ⁢                                                              /                                                            ⁢                                                              T                                s                                                                                                              )                                                ×                        k                                            )                                                                                            +                          v              k                                                                    =                    ⁢                                    h              ×                              exp                ⁡                                  (                                                            -                      j                                        ×                                                                  2                        ⁢                        π                                            N                                        ×                    k                    ×                    Δ                    ⁢                                                                                  ⁢                    t                    ⁢                                          /                                        ⁢                                          T                      s                                                        )                                            ×                                                ∑                                      n                    =                    0                                                        N                    -                    1                                                  ⁢                                                      x                    n                                    ⁢                                      exp                    ⁡                                          (                                                                        -                          j                                                ×                                                                              2                            ⁢                            π                                                    N                                                ×                        n                        ×                        k                                            )                                                                                            +                          v              k                                          Then it can be concluded that
            r      k        =                  h        ×                  exp          ⁡                      (                                          -                j                            ×                                                2                  ⁢                  π                                N                            ×              k              ×              Δ              ⁢                                                          ⁢              t              ⁢                              /                            ⁢                              T                s                                      )                          ×                  y          k                    +              v        k              ;wherein N is the number of sub-carriers; h is the channel response (phase and amplitude) difference to the reference antenna/TX (RX) path; Δt is the time delay difference relative to the reference antenna; and Ts is the interval of samples.
Thus the phase shift of k-th sub-carrier in the frequency domain is:
            θ      k        =                            -                                    2              ⁢              π                        N                          ×        k        ×        Δ        ⁢                                  ⁢        t        ⁢                  /                ⁢                  T          s                    +              φ                  h          ,          k                    +              ɛ        k              ;wherein φh,k is the phase shift caused by h, which is constant in the whole working frequency band as designed; and εk is the phase shift caused by the noise. According to the signal processing theory, k=0 represents the middle frequency (carrier frequency) in the signal band; k=N/2−1 represents the lowest frequency; and k=N/2 represents the highest frequency in the signal frequency band.
Therefore, it can be concluded that the delay difference will cause a larger phase shift in the higher/lower frequency band, which will reduce the beamforming gain. Consequently, it is not only necessary to estimate the delay difference in frequency domain, but also necessary to compensate for this delay difference. There also exists need to improve the accuracy in the delay difference estimation.